# Long range scattering for the nonlinear Schr"odinger equation with higher order anisotropic dispersion in two dimensions

## TL;DR

This paper proves long range scattering for a two-dimensional nonlinear Schrödinger equation with higher order anisotropic dispersion, demonstrating solutions that asymptotically approach a specified profile with a logarithmic phase correction.

## Contribution

It establishes the long range scattering for 2D higher order anisotropic NLS with quadratic nonlinearity, extending previous studies to include this specific case.

## Key findings

- Constructed solutions converging to prescribed asymptotic profiles.
- Demonstrated the necessity of logarithmic phase correction.
- Extended understanding of long time behavior in anisotropic dispersive equations.

## Abstract

This paper is a continuation of our previous study on the long time behavior of solution to the nonlinear Schr"odinger equation with higher order anisotropic dispersion (4NLS). We prove the long range scattering for (4NLS) with the quadratic nonlinearity in two dimensions. More precisely, for a given asymptotic profile $u_{+}$, we construct a solution to (4NLS) which converges to $u_{+}$ as t to infinity, where $u_{+}$ is given by the leading term of the solution to the linearized equation of (4NLS) with a logarithmic phase correction.